few_shots: Example:
Problem/activity:
Narrative
The purpose of this activity is for students to create composite shapes and compose new shapes from composite shapes. For example, students may find that a rectangular prism can be made from four triangular prisms or from eight smaller cubes. Students then discover that they can combine two of the composed rectangular prisms to make a new rectangular prism.
Required Materials
Materials to Gather
Geoblocks
Launch
Groups of 4
Give each group geoblocks.
Display a large cube.
“This cube can be built from other shapes. See how many ways you can build a cube using other shapes.”
3 minutes: partner work time
Share responses.
Activity
“Choose a different shape. Try to find smaller shapes you can use to build that shape.”
5 minutes: partner work time
Student Response
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Activity Synthesis
Display a cube and a composite cube—for example, a cube composed of four small cubes and one rectangular prism.
Demonstrate putting the shapes together.
“What new shape did I make with the cube and the cube I made?”
“Put your shapes together to make a rectangular prism.”
Display a rectangular prism and two triangular prisms.
“Find these shapes. Put them together. What new shape can you make?” (a cube, a rectangular block)


Mathematical concept/skill options: 
A. Reason with shapes and their attributes.
B. Understand similarity in terms of similarity transformations.
C. Use coordinates to prove simple geometric theorems algebraically.
D. Translate between the geometric description and the equation for a conic section.
E. Understand congruence and similarity using physical models, transparencies, or geometry software.
F. Analyze, compare, create, and compose shapes.
G. Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
H. Understand congruence in terms of rigid motions.
I. Classify two-dimensional figures into categories based on their properties.
J. Draw, construct, and describe geometrical figures and describe the relationships between them.
K. Explain volume formulas and use them to solve problems.
L. Prove theorems involving similarity.
M. Prove geometric theorems.
N. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
O. Define trigonometric ratios and solve problems involving right triangles.
P. Apply trigonometry to general triangles.
Q. Visualize relationships between two-dimensional and three-dimensional objects.
R. Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
S. Make geometric constructions.
T. Apply geometric concepts in modeling situations.
U. Solve real-world and mathematical problems involving area, surface area, and volume.
V. Graph points on the coordinate plane to solve real-world and mathematical problems.
W. Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
X. Understand and apply theorems about circles.
Y. Understand and apply the Pythagorean Theorem.
Z. Experiment with transformations in the plane.
AA. Find arc lengths and areas of sectors of circles.


Thought: This activity asks students to reason how shapes can combine to make other shapes.
Answer: A

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_lesson_009431

1
options: Analyze proportional relationships and use them to solve real-world and mathematical problems.

Understand ratio concepts and use ratio reasoning to solve problems.
problem_activity: Activity
In this task, students calculate and interpret both
\(\frac{a}{b}\)
and
\(\frac{b}{a}\)
from a ratio
\(a:b\)
presented in a context. They work with less-familiar units. The term
unit rate
is introduced so that students have a general name for a “how many per 1” quantity.
In the first half of the task, students practice computing unit rates from ratios. In the second half they practice selecting the better unit rate to use (
\(\frac{a}{b}\)
or
\(\frac{b}{a}\)
) based on the question posed.
As students work on the second half of the task, identify 1–2 students per question to share their choice of unit rate and how it was used to answer the question.
Launch
Recap that in the previous activity the ratio of 15 cups water for every 6 cups oats can be expressed as two rates “per 1”. These rates are 0.4 cups of oats per cup of water or 2.5 cups of water per cup of oats. Emphasize that, in a table, each of these rates reflects a value paired with a “1” in a row, and that both can be useful depending on the problem at hand. Tell students that we call 0.4 and 2.5 “unit rates” and that a unit rate means “the amount of one quantity for 1 of another quantity.”
Arrange students in groups of 2. Tell students that they will now solve some problems using unit rates. Give students 3–4 minutes to complete the first half of the task (the first three problems). Ask them to share their responses with their partner and come to an agreement before moving on to the second half. Clarify that “oz” is an abbreviation for “ounce.”
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of these terms. Include the following term and maintain the display for reference throughout the unit: unit rate.
Supports accessibility for: Conceptual processing; Language
Writing, Listening: MLR6 Three Reads.
Use this routine to set students up to comprehend each situation, without solving the questions for them. In the first read, students read the situation with the goal of comprehending the situation (e.g., a cheesecake recipe contains cream cheese and sugar, Mai’s family drinks milk regularly, Tyler has raffle tickets). In the second read, ask students to identify the important quantities by asking them what can be counted or measured (e.g., 12 oz of cream cheese with 15 oz of sugar, 10 gallons of milk every 6 weeks, $16 for 4 raffle tickets). In the third read, ask students to brainstorm possible mathematical solution strategies to answer the question. This helps students connect the ratio language used in the problem to identify a strategy for finding a unit rate.
Design Principle(s): Support sense-making
Student Facing
For each situation, find the
unit rates
.
A cheesecake recipe says, “Mix 12 oz of cream cheese with 15 oz of sugar.”
How many ounces of cream cheese are there for every ounce of sugar?
How many ounces of sugar is that for every ounce of cream cheese?
Mai’s family drinks a total of 10 gallons of milk every 6 weeks.
How many gallons of milk does the family drink per week?
How many weeks does it take the family to consume 1 gallon of milk?
Tyler paid
$
16 for 4 raffle tickets.
What is the price per ticket?
How many tickets is that per dollar?
For each problem, decide which unit rate from the previous situations you prefer to use. Next, solve the problem, and show your thinking.
If Lin wants to make extra cheesecake filling, how much cream cheese will she need to mix with 35 ounces of sugar?
How many weeks will it take Mai’s family to finish 3 gallons of milk?
How much would all 1,000 raffle tickets cost?
Student Response
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Student Facing
Are you ready for more?
Write a “deal” on tickets for Tyler’s raffle that sounds good, but is actually a little worse than just buying tickets at the normal price.
Student Response
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Anticipated Misconceptions
If students are not sure how to use the unit rates they found for each situation to answer the second half of the task, remind them of how the oatmeal problem was solved. Suggest that this problem is similar because they can scale up from a unit rate to answer the questions.
Activity Synthesis
Invite previously identified students to share their work on the second half (the last three questions) of the task.
Though the task prompts students to think in terms of unit rate, some students may still reason in ways that feel safer. For example, to find out how much cream cheese Lin would mix with 35 oz of sugar, they may double the 12 oz of cream cheese to 15 oz of sugar ratio to obtain 24 oz of cream cheese for 30 oz of sugar, and then add 4 oz more of cream cheese for the additional 5 oz of sugar. Such lines of reasoning show depth of understanding and should be celebrated. Guide students to also see, however, that some problems (such as the milk problem) can be more efficiently solved using unit rates.
For the ticket problem, students may comment that
\(\frac14\)
of a ticket costing a dollar does not make sense, since it is not possible to purchase
\(\frac14\)
of a ticket. Take this opportunity to applaud the student(s) for reasoning about the interpretation of the number in the context, which is an example of engaging in MP2. If students do not raise this concern, ask: “How can
\(\frac14\)
of a ticket costing a dollar make sense?” Students may argue that the quantity, on its own, does not make sense. Challenge them to figure out how the rate could be used in the context of the problem. For example, ask, “If I had
$
80, how many tickets could I buy? What if I had
$
75? Can the ‘
\(\frac14\)
of a ticket per dollar’ rate help answer these questions?”


correct_option_index: 1